Precalculus Examples

$\frac{5 x^{2} + 10 x + 13}{(x + 2) {(x + 6)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{x + 2}$

Step 1.2

$\frac{A}{x + 2} + \frac{B}{{(x + 6)}^{2}}$

Step 1.3

$\frac{A}{x + 2} + \frac{B}{{(x + 6)}^{2}} + \frac{C}{x + 6}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 2) {(x + 6)}^{2}$ .

$\frac{(5 x^{2} + 10 x + 13) (x + 2) {(x + 6)}^{2}}{(x + 2) {(x + 6)}^{2}} = \frac{(A) (x + 2) {(x + 6)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.5

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

Step 1.5.2

Rewrite the expression.

$\frac{(5 x^{2} + 10 x + 13) {(x + 6)}^{2}}{{(x + 6)}^{2}} = \frac{(A) (x + 2) {(x + 6)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.6

Cancel the common factor of ${(x + 6)}^{2}$ .

Tap for more steps...

Step 1.6.1

Cancel the common factor.

$\frac{(5 x^{2} + 10 x + 13) {(x + 6)}^{2}}{{(x + 6)}^{2}} = \frac{(A) (x + 2) {(x + 6)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.6.2

Divide $5 x^{2} + 10 x + 13$ by $1$ .

$5 x^{2} + 10 x + 13 = \frac{(A) (x + 2) {(x + 6)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7

Simplify each term.

Tap for more steps...

Step 1.7.1

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 1.7.1.1

Cancel the common factor.

$5 x^{2} + 10 x + 13 = \frac{A (x + 2) {(x + 6)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.1.2

Divide $(A) {(x + 6)}^{2}$ by $1$ .

$5 x^{2} + 10 x + 13 = (A) {(x + 6)}^{2} + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.2

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$5 x^{2} + 10 x + 13 = A ((x + 6) (x + 6)) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.3

Expand $(x + 6) (x + 6)$ using the FOIL Method.

Tap for more steps...

Step 1.7.3.1

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A (x (x + 6) + 6 (x + 6)) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.3.2

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A (x \cdot x + x \cdot 6 + 6 (x + 6)) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.3.3

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.4

Simplify and combine like terms.

Tap for more steps...

Step 1.7.4.1

Simplify each term.

Tap for more steps...

Step 1.7.4.1.1

Multiply $x$ by $x$ .

$5 x^{2} + 10 x + 13 = A (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.4.1.2

Move $6$ to the left of $x$ .

$5 x^{2} + 10 x + 13 = A (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.4.1.3

Multiply $6$ by $6$ .

$5 x^{2} + 10 x + 13 = A (x^{2} + 6 x + 6 x + 36) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.4.2

Add $6 x$ and $6 x$ .

$5 x^{2} + 10 x + 13 = A (x^{2} + 12 x + 36) + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.5

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + A (12 x) + A \cdot 36 + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.6

Simplify.

Tap for more steps...

Step 1.7.6.1

Rewrite using the commutative property of multiplication.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + A \cdot 36 + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.6.2

Move $36$ to the left of $A$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 \cdot A + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.7

Cancel the common factor of ${(x + 6)}^{2}$ .

Tap for more steps...

Step 1.7.7.1

Cancel the common factor.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + \frac{(B) (x + 2) {(x + 6)}^{2}}{{(x + 6)}^{2}} + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.7.2

Divide $(B) (x + 2)$ by $1$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + (B) (x + 2) + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.8

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + B \cdot 2 + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.9

Move $2$ to the left of $B$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 \cdot B + \frac{(C) (x + 2) {(x + 6)}^{2}}{x + 6}$

Step 1.7.10

Cancel the common factor of ${(x + 6)}^{2}$ and $x + 6$ .

Tap for more steps...

Step 1.7.10.1

Factor $x + 6$ out of $(C) (x + 2) {(x + 6)}^{2}$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + \frac{(x + 6) ((C) (x + 2) (x + 6))}{x + 6}$

Step 1.7.10.2

Cancel the common factors.

Tap for more steps...

Step 1.7.10.2.1

Multiply by $1$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + \frac{(x + 6) ((C) (x + 2) (x + 6))}{(x + 6) \cdot 1}$

Step 1.7.10.2.2

Cancel the common factor.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + \frac{(x + 6) ((C) (x + 2) (x + 6))}{(x + 6) \cdot 1}$

Step 1.7.10.2.3

Rewrite the expression.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + \frac{(C) (x + 2) (x + 6)}{1}$

Step 1.7.10.2.4

Divide $(C) (x + 2) (x + 6)$ by $1$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + (C) (x + 2) (x + 6)$

Step 1.7.11

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + (C x + C \cdot 2) (x + 6)$

Step 1.7.12

Move $2$ to the left of $C$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + (C x + 2 \cdot C) (x + 6)$

Step 1.7.13

Expand $(C x + 2 C) (x + 6)$ using the FOIL Method.

Tap for more steps...

Step 1.7.13.1

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x (x + 6) + 2 C (x + 6)$

Step 1.7.13.2

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x \cdot x + C x \cdot 6 + 2 C (x + 6)$

Step 1.7.13.3

Apply the distributive property.

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x \cdot x + C x \cdot 6 + 2 C x + 2 C \cdot 6$

Step 1.7.14

Simplify and combine like terms.

Tap for more steps...

Step 1.7.14.1

Simplify each term.

Tap for more steps...

Step 1.7.14.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.7.14.1.1.1

Move $x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C (x \cdot x) + C x \cdot 6 + 2 C x + 2 C \cdot 6$

Step 1.7.14.1.1.2

Multiply $x$ by $x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x^{2} + C x \cdot 6 + 2 C x + 2 C \cdot 6$

Step 1.7.14.1.2

Move $6$ to the left of $C x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x^{2} + 6 \cdot (C x) + 2 C x + 2 C \cdot 6$

Step 1.7.14.1.3

Multiply $6$ by $2$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x^{2} + 6 C x + 2 C x + 12 C$

Step 1.7.14.2

Add $6 C x$ and $2 C x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 A x + 36 A + B x + 2 B + C x^{2} + 8 C x + 12 C$

Step 1.8

Simplify the expression.

Tap for more steps...

Step 1.8.1

Move $A$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 x A + 36 A + B x + 2 B + C x^{2} + 8 C x + 12 C$

Step 1.8.2

Reorder $B$ and $x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 x A + 36 A + B x + 2 B + C x^{2} + 8 C x + 12 C$

Step 1.8.3

Move $2 B$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 x A + 36 A + B x + C x^{2} + 8 C x + 2 B + 12 C$

Step 1.8.4

Move $36 A$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 x A + B x + C x^{2} + 8 C x + 36 A + 2 B + 12 C$

Step 1.8.5

Move $B x$ .

$5 x^{2} + 10 x + 13 = A x^{2} + 12 x A + C x^{2} + B x + 8 C x + 36 A + 2 B + 12 C$

Step 1.8.6

Move $12 x A$ .

$5 x^{2} + 10 x + 13 = A x^{2} + C x^{2} + 12 x A + B x + 8 C x + 36 A + 2 B + 12 C$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$5 = A + C$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$10 = 12 A + B + 8 C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$13 = 36 A + 2 B + 12 C$

Step 2.4

Set up the system of equations to find the coefficients of the partial fractions.

$5 = A + C$

$10 = 12 A + B + 8 C$

$13 = 36 A + 2 B + 12 C$

$5 = A + C$

$10 = 12 A + B + 8 C$

$13 = 36 A + 2 B + 12 C$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Solve for $A$ in $5 = A + C$ .

Tap for more steps...

Step 3.1.1

Rewrite the equation as $A + C = 5$ .

$A + C = 5$

$10 = 12 A + B + 8 C$

$13 = 36 A + 2 B + 12 C$

Step 3.1.2

Subtract $C$ from both sides of the equation.

$A = 5 - C$

$10 = 12 A + B + 8 C$

$13 = 36 A + 2 B + 12 C$

$A = 5 - C$

$10 = 12 A + B + 8 C$

$13 = 36 A + 2 B + 12 C$

Step 3.2

Replace all occurrences of $A$ with $5 - C$ in each equation.

Tap for more steps...

Step 3.2.1

Replace all occurrences of $A$ in $10 = 12 A + B + 8 C$ with $5 - C$ .

$10 = 12 (5 - C) + B + 8 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $12 (5 - C) + B + 8 C$ .

Tap for more steps...

Step 3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1.1

Apply the distributive property.

$10 = 12 \cdot 5 + 12 (- C) + B + 8 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

Step 3.2.2.1.1.2

Multiply $12$ by $5$ .

$10 = 60 + 12 (- C) + B + 8 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

Step 3.2.2.1.1.3

Multiply $- 1$ by $12$ .

$10 = 60 - 12 C + B + 8 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

$10 = 60 - 12 C + B + 8 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

Step 3.2.2.1.2

Add $- 12 C$ and $8 C$ .

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 36 A + 2 B + 12 C$

Step 3.2.3

Replace all occurrences of $A$ in $13 = 36 A + 2 B + 12 C$ with $5 - C$ .

$13 = 36 (5 - C) + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

Step 3.2.4

Simplify the right side.

Tap for more steps...

Step 3.2.4.1

Simplify $36 (5 - C) + 2 B + 12 C$ .

Tap for more steps...

Step 3.2.4.1.1

Simplify each term.

Tap for more steps...

Step 3.2.4.1.1.1

Apply the distributive property.

$13 = 36 \cdot 5 + 36 (- C) + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

Step 3.2.4.1.1.2

Multiply $36$ by $5$ .

$13 = 180 + 36 (- C) + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

Step 3.2.4.1.1.3

Multiply $- 1$ by $36$ .

$13 = 180 - 36 C + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 180 - 36 C + 2 B + 12 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

Step 3.2.4.1.2

Add $- 36 C$ and $12 C$ .

$13 = 180 + 2 B - 24 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 180 + 2 B - 24 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 180 + 2 B - 24 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

$13 = 180 + 2 B - 24 C$

$10 = 60 + B - 4 C$

$A = 5 - C$

Step 3.3

Reorder $5$ and $- C$ .

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

$10 = 60 + B - 4 C$

Step 3.4

Solve for $B$ in $10 = 60 + B - 4 C$ .

Tap for more steps...

Step 3.4.1

Rewrite the equation as $60 + B - 4 C = 10$ .

$60 + B - 4 C = 10$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

Step 3.4.2

Move all terms not containing $B$ to the right side of the equation.

Tap for more steps...

Step 3.4.2.1

Subtract $60$ from both sides of the equation.

$B - 4 C = 10 - 60$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

Step 3.4.2.2

Add $4 C$ to both sides of the equation.

$B = 10 - 60 + 4 C$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

Step 3.4.2.3

Subtract $60$ from $10$ .

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 180 + 2 B - 24 C$

Step 3.5

Replace all occurrences of $B$ with $- 50 + 4 C$ in each equation.

Tap for more steps...

Step 3.5.1

Replace all occurrences of $B$ in $13 = 180 + 2 B - 24 C$ with $- 50 + 4 C$ .

$13 = 180 + 2 (- 50 + 4 C) - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify $180 + 2 (- 50 + 4 C) - 24 C$ .

Tap for more steps...

Step 3.5.2.1.1

Simplify each term.

Tap for more steps...

Step 3.5.2.1.1.1

Apply the distributive property.

$13 = 180 + 2 \cdot - 50 + 2 (4 C) - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.5.2.1.1.2

Multiply $2$ by $- 50$ .

$13 = 180 - 100 + 2 (4 C) - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.5.2.1.1.3

Multiply $4$ by $2$ .

$13 = 180 - 100 + 8 C - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 180 - 100 + 8 C - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.5.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 3.5.2.1.2.1

Subtract $100$ from $180$ .

$13 = 80 + 8 C - 24 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.5.2.1.2.2

Subtract $24 C$ from $8 C$ .

$13 = 80 - 16 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 80 - 16 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 80 - 16 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 80 - 16 C$

$B = - 50 + 4 C$

$A = - C + 5$

$13 = 80 - 16 C$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6

Solve for $C$ in $13 = 80 - 16 C$ .

Tap for more steps...

Step 3.6.1

Rewrite the equation as $80 - 16 C = 13$ .

$80 - 16 C = 13$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.2

Move all terms not containing $C$ to the right side of the equation.

Tap for more steps...

Step 3.6.2.1

Subtract $80$ from both sides of the equation.

$- 16 C = 13 - 80$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.2.2

Subtract $80$ from $13$ .

$- 16 C = - 67$

$B = - 50 + 4 C$

$A = - C + 5$

$- 16 C = - 67$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.3

Divide each term in $- 16 C = - 67$ by $- 16$ and simplify.

Tap for more steps...

Step 3.6.3.1

Divide each term in $- 16 C = - 67$ by $- 16$ .

$\frac{- 16 C}{- 16} = \frac{- 67}{- 16}$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.3.2

Simplify the left side.

Tap for more steps...

Step 3.6.3.2.1

Cancel the common factor of $- 16$ .

Tap for more steps...

Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{- 16 C}{- 16} = \frac{- 67}{- 16}$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 67}{- 16}$

$B = - 50 + 4 C$

$A = - C + 5$

$C = \frac{- 67}{- 16}$

$B = - 50 + 4 C$

$A = - C + 5$

$C = \frac{- 67}{- 16}$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.6.3.3

Simplify the right side.

Tap for more steps...

Step 3.6.3.3.1

Dividing two negative values results in a positive value.

$C = \frac{67}{16}$

$B = - 50 + 4 C$

$A = - C + 5$

$C = \frac{67}{16}$

$B = - 50 + 4 C$

$A = - C + 5$

$C = \frac{67}{16}$

$B = - 50 + 4 C$

$A = - C + 5$

$C = \frac{67}{16}$

$B = - 50 + 4 C$

$A = - C + 5$

Step 3.7

Replace all occurrences of $C$ with $\frac{67}{16}$ in each equation.

Tap for more steps...

Step 3.7.1

Replace all occurrences of $C$ in $B = - 50 + 4 C$ with $\frac{67}{16}$ .

$B = - 50 + 4 (\frac{67}{16})$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2

Simplify the right side.

Tap for more steps...

Step 3.7.2.1

Simplify $- 50 + 4 (\frac{67}{16})$ .

Tap for more steps...

Step 3.7.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.7.2.1.1.1

Factor $4$ out of $16$ .

$B = - 50 + 4 (\frac{67}{4 (4)})$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.1.2

Cancel the common factor.

$B = - 50 + 4 (\frac{67}{4 \cdot 4})$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.1.3

Rewrite the expression.

$B = - 50 + \frac{67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

$B = - 50 + \frac{67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.2

To write $- 50$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$B = - 50 \cdot \frac{4}{4} + \frac{67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.3

Combine $- 50$ and $\frac{4}{4}$ .

$B = \frac{- 50 \cdot 4}{4} + \frac{67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.4

Combine the numerators over the common denominator.

$B = \frac{- 50 \cdot 4 + 67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.5

Simplify the numerator.

Tap for more steps...

Step 3.7.2.1.5.1

Multiply $- 50$ by $4$ .

$B = \frac{- 200 + 67}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.5.2

Add $- 200$ and $67$ .

$B = \frac{- 133}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

$B = \frac{- 133}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.2.1.6

Move the negative in front of the fraction.

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = - C + 5$

Step 3.7.3

Replace all occurrences of $C$ in $A = - C + 5$ with $\frac{67}{16}$ .

$A = - (\frac{67}{16}) + 5$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.7.4

Simplify the right side.

Tap for more steps...

Step 3.7.4.1

Simplify $- (\frac{67}{16}) + 5$ .

Tap for more steps...

Step 3.7.4.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$A = - \frac{67}{16} + 5 \cdot \frac{16}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.7.4.1.2

Combine $5$ and $\frac{16}{16}$ .

$A = - \frac{67}{16} + \frac{5 \cdot 16}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.7.4.1.3

Combine the numerators over the common denominator.

$A = \frac{- 67 + 5 \cdot 16}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.7.4.1.4

Simplify the numerator.

Tap for more steps...

Step 3.7.4.1.4.1

Multiply $5$ by $16$ .

$A = \frac{- 67 + 80}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.7.4.1.4.2

Add $- 67$ and $80$ .

$A = \frac{13}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = \frac{13}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = \frac{13}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = \frac{13}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

$A = \frac{13}{16}$

$B = - \frac{133}{4}$

$C = \frac{67}{16}$

Step 3.8

List all of the solutions.

$A = \frac{13}{16}, B = - \frac{133}{4}, C = \frac{67}{16}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 2} + \frac{B}{{(x + 6)}^{2}} + \frac{C}{x + 6}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{\frac{13}{16}}{x + 2} + \frac{- \frac{133}{4}}{{(x + 6)}^{2}} + \frac{\frac{67}{16}}{x + 6}$